Hann function: Difference between revisions

The '''Hann function''' ofis lengthnamed <math>L,</math>after the Austrian meteorologist [[Julius von Hann]]. It is a [[window function]] used to perform '''Hann smoothing''',.<ref>{{Cite book|urlname=http:Essenwanger//worldcat.org/oclc/152410575|title=Elements> ofThe statistical analysis|last=Essenwangerfunction, O.with M.length (Oskar M.)|date=1986|publisher=Elsevier|isbn=0444424261|oclc=152410575}}<math>L</refmath> isand namedamplitude after the Austrian meteorologist [[Julius von Hann]]<math>1/L,</math> is a [[window function]] given by''':'''

\tfrac{1}{2L}\left(\tfrac{1}{2} + \tfrac{1}{2} \cos \left(\frac{2\pi x}{L} \right) \right) = \tfrac{1}{L}\cos^2 \left(\frac{\pi x}{L}\right),\quad &\left|x\right| \leq L/2\\

For [[digital signal processing]], the function can beis sampled symmetrically as(with spacing <math>L/N</math> and amplitude <math>1</math>)''':'''

w[n] = L\cdot w_0\left(\tfrac{L}{N} (n-N/2)\right) &= \tfrac{1}{2} \left[1 - \cos \left ( \tfrac{2 \pi n}{N} \right) \right]\\

[[File: DFT-even Hann window & spectral leakage.png|thumb|300px|right|Top: 16 sample [[Window_functionSpectral_leakage#DFT-evensymmetry|DFT-even]] Hann window. Bottom: Its discrete-time Fourier transform (DTFT) and the 3 non-zero values of its discrete Fourier transform (DFT).]]

The [[Discrete-time Fourier transform]] (DTFT) of the <math>N+1</math> length, time-shifted sequence is defined by a Fourier series, which also has a 3-term equivalent that is derived similarly to the Fourier transform derivation''':'''

For even values of N, theThe truncated sequence <math>\{w[n],\ 0 \le n \le N-1\}</math> is a [[Window_functionSpectral_leakage#DFT-evensymmetry|DFT-even]] (aka ''periodic'') Hann window. Since the truncated sample has value zero, it is clear from the Fourier series definition that the DTFTs are equivalent. However, the approach followed above results in a significantly different-looking, but equivalent, 3-term expression''':'''

An ''N''-length DFT of the window function samples the DTFT at frequencies <math>f = k/N,</math> for integer values of <math>k.</math> From the expression immediately above, it is easy to see that only 3 of the N DFT coefficients are non-zero. And from the other expression, it is apparent that all are real-valued. These properties are appealing for real-time applications that require both windowed and non-windowed (rectangularly windowed) transforms, because the windowed transforms can be efficiently derived from the non-windowed transforms by [[Discrete Fourier transform#Convolution theorem duality|convolution]].<ref>{{cite patent |ref=refCarlin |inventor-last name=Carlin |inventor-first=Joe |inventor2-last=Collins |inventor2-first=Terry |inventor3-last=Hays |inventor3-first=Peter |inventor4-last=Hemmerdinger |inventor4-first=Barry |inventor5-last=Kellogg |inventor5-first=Robert |inventor6-last=Kettig |inventor6-first=Robert |inventor7-last=Lemmon |inventor7-first=Bradley |inventor8-last=Murdock |inventor8-first=Thomas |inventor9-last=Tamaru |inventor9-first=Robert |inventor10-last=Ware |inventor10-first=Stuart |date=1999 |issue-date=2005 |title=Wideband communication intercept and direction finding device using hyperchannelization |country-code=US |description=patent |patent-number=6898235}}</ref>{{efn-la

The function is named in honourhonor of von Hann, who used the three-term weighted average smoothing technique on meteorological data.<ref>{{Cite book|url=https://books.google.com/books?id=Ma3PAAAAMAAJ&pg=PA199|title=Handbook of Climatology|lastname=Hann|first=Julius von|date=1903|publisher=Macmillan|year=|isbn=|location=|pages=|language=en|quote=The figures under ''b'' are determined by taking into account the parallels 5° away on either side. Thus, for example, for latitude 60° we have ½[60+(65+55)÷2].}}</ref><ref name=":1" Kahlig/> However, the erroneous<refterm name=":0" /> "''Hanning"'' function is also heardconventionally ofused,<ref on occasion,name=Harris/> derived from the paper in which it was named, where the term "''hanning a signal"'' was used to mean applying the Hann window to it.<ref>{{Cite journal|lastname=Blackman|first=R. B.|last2=Tukey|first2=J. W.|date=1958|title=The measurement of power spectra from the point of view of communications engineering — Part I|url=https://ieeexplore.ieee.org/document/6768513/|journal=The Bell System Technical Journal|volume=37|issue=1|pages=273|doi=10.1002/j.1538-7305.1958.tb03874.x|issn=0005-8580|via=}}</ref><ref>{{Cite book|urlname=https:Blackman2//archive.org/details/TheMeasurementOfPowerSpectra/page/n58|title=The measurement of power spectra from the point of view of communications engineering|last=Blackman|first=R. B. (Ralph Beebe)|last2=Tukey|first2=John W. (John Wilder)|date=1959|publisher=New York : Dover Publications|year=|isbn=|location=|pages=98|lccn=59-10185}}</ref> The confusion arose from the similar [[Hamming function]], named after [[Richard Hamming]].